The appearance of the projection \(f\mapsto \int f(y)\, dy\)  onto the constant functions is natural in view of the fact that
the resonant function of \(-\partial _x^2\)  at zero energy is \(f\equiv 1\) .
This also shows that one should expect \(t^{-\frac{3}{2}}\)  local decay for one-dimensional operators without zero energy resonance (note that, however, the global decay as in () is never faster than \(t^{-\frac{1}{2}}\)  if \(d=1\) ), at least assuming sufficient decay of \(V\) .
This is indeed the case, see [1]}. In two dimensions, Murata obtained the faster local \(L^2({\mathbb {R}}^2)\)  decay \(t^{-1}\log ^{-2} t\)  for
operators without resonance. Erdogan, Green [2]} established the more difficult sharp weighted \(L^1\rightarrow L^\infty \)  version of these global bounds in \({\mathbb {R}}^2\)  assuming that 0 energy is regular. These faster local decays (as compared to the global \(L^\infty \)  decay) play a crucial role in certain
applications to nonlinear stability results, see Buslaev, Perelman [3]}, Krieger, Schlag [4]} for the
one-dimensional case, and Kirr, Zarnescu [5]} for examples of two-dimensional applications. Loosely speaking, the point here is that in contrast
to the global decay rates these faster non-resonant local rates are integrable in time which allows one to close certain bootstrap
arguments involving the Duhamel formula.
